This is only a preliminary introductory answer to shed some light on the question and clarify a proposed answer.

The exact Shrodinger equation SE was invented by Schrodinger to describe the probability amplitude of the wave function in the infinite space xyz supplemented by the real time t as an external effect.

It cannot handle closed systems bounded by Dirichlet boundary conditions such as the time-dependent 2D and 3D heat diffusion equation.

Dirichlet boundary conditions distinguish closed/insulated systems from open systems as in the case of closed set and open set in mathematics.

A closed set and a closed system can be defined as those systems where the boundaries are part of the system itself.

Note that this is the case for chains of transition matrices B where the Dirichlet BC for closed systems implies that time is woven into xyz space as a unit block of 4D x-t space.

The question arises whether an SE can be modified to correctly describe a bounded system?

We assume the answer is yes and that quantum mechanics and hyper-classical physics fit together.

First of all h does not change

Then se is after bohr it improves it

The iron guards of the Bohr/Copenhagen interpretation defend to the last breath all issues of this interpretation, whether true or false.

Thanks to the opposition of the new generation of theoretical physicists and mathematicians, the old iron guards are only a small remnant of the era of MacArthism where the mastery of quantum mechanics dominated: Shut up and calculate.

Going back to SE with Bohr's interpretation, I guess these are the three most common mistakes while I'm sure most Researchgate contributors can find more:

1- Units and Dimensions

Bohr/Copenhagen interpretation of the wave function Ψ

defines the units and dimensions of Ψ differently depending on the number of dimensions of the receiving space.

i-For a spatial wave function of position 1D Ψ(x)

the normalization condition would be ∫Ψ∗(x)Ψ(x)dx=1

so Ψ has inverse square root units of length, which is m−1/2

ii-Similarly, for a three-dimensional position, the spatial wave function Ψ(x) has its own normalization condition such as:

∫Ψ∗(x)Ψ(x)d^3x=1

Ψ(x) has units of square root of inverse volume, which is m−3/2

Moreover, theoretical physics does not allow it, it is not acceptable.

Units and dimensions must be raised to zero or a whole whole power, but never to a fraction.

* If we compare this to the case of replacing SE with the statistical matrices Q and W, we find that the dimensions of W and Q are always the square root of the energy density per unit volume and the same is true for 1D, 2D and 3D.

The normalization condition is a pure summation condition and not an integration condition which is the sum of the multiplication.

.

2- The problem of distance and infinity

Distance is still defined in essentially the same way as in classical physics. In other words, spacetime is a flat Euclidean space with the usual Euclidean distance.

If we talk about the distance between two quantum particles, they must have a definite position in order to measure the distance between them. This requires them to be in proper positional states.

We assume that the SE interpretation with Bohr is limited to the study of the hydrogen-like atom where the total diameter distance is a few angstroms (Å) (1 A = 1E-10 m) and the effective electron radius is estimated at E -2 angstrom (Å).

The n=infinity mentioned in SE relates to the quantum number of infinite principle n which again corresponds to the diameter of the hydrogen atom by a few angeströms.

Obviously, the units and dimensions of the distance are not well defined in the SE interpretation.

** If we compare this to the case of replacing SE by the statistical matrices Q and W, we see that in the latter case the distance is well defined in 1D, 2D and 3D geometric spaces simply by ordering and arranging the nodes in these as we did for the actual transition matrix B.

3- Rigorous derivation

There is no rigorous or even serious derivation of SE with the Bohr/Copenhagen interpretation.

E. Schrödinger's solid derivation in 1927 proved other concepts in his equation and he himself considered the Bohr/Copenhagen interpretation an unpleasant joke. He expressed his feelings and thoughts through his well-known paradox. (Schrödinger's cat dead or alive).

*** If we compare this to the case of replacing SE with the numerical solution of the temporal chains of the statistical matrix Q, we see that in the latter case the derivation emerges from the well-proven statistical transition matrix B. Moreover, the results of the B-strings and Q-strings are

numerically validated.

To be continued.

Once again the Bohr method is already superceded by Schrodinger.

So you talk about old stuff

So give your explanation...

In many cases there is no interpretation

Beyond energy levels given by

Eigenvalues.

You're stirring together issues from various fields, not all of which are affiliated to the Copenhagen interpretation. The normalization conditions for the wavefunction are also a requirement in a non-Copenhagen world. At least non of the alternative concepts I've seen so far (most of which I learned about from Friebe's Philosophy of Quantum Mechanics) would work without a normalizeable wavefunction.

If you have experimental findings which suggest a non-normalizable wavefunction would be physically valid, please feel free to share them.

Answer II-Continued.

What is waving in the Schrödinger equation and why is it called the “wave” of the Schrödinger equation, especially when its phase is undefined?

It is well known that there is theoretical and experimental evidence for a causal relationship between the phase of the wave function and physical reality.

The Copenhagen interpretation of quantum mechanics, which only gives physical meaning to the magnitude of the wave function, cannot be considered complete on this basis.

* A new dynamic-statistical interpretation of quantum mechanics is needed [1,2].

Believe it or not, attaching a well-defined phase to the amplitude of the SE wave would no longer complicate it but on the contrary would make it more understandable and its solution more accessible.

However, we assume that defining a phase at the amplitude of SE can be done via two different approaches:

i-reform the Bohr/Copenhagen interpretation of the Schrödinger equation.

ii-Apply the complex transition matrix Q to find the statistical numerical solution of SE.

To be continued.

1-Ivan Georgiev Koprinkov, Phase Causation of the Wave Function or Can the Copenhagen Interpretation of Quantum Mechanics Be Considered Complete? Journal of Modern Physics Vol.7 No.4, February 2016.

2-I.Abbas,Numerical statistical resolution of the Schrödinger wave equation, Researchgate.

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1) I actually tried finding your reference 2, when I search your research items for "Schrödinger" I get 0 results.

2) The phase is not undefined and it matters in the interpretation of a lot of spectroscopies, especially those which are spin-related. All NMR pulse sequences are based on relative phases at different points of time. The only thing which is undefined is the initial value.

Ismail Abbas - re "the exact Shrodinger equation"

I wouldn't describe the Schrodinger equation as "exact" - it is in fact the mass-dominated limit of the Klein-Gordon equation. There are various ways to derive the SE from the KGE, but my favourite (:) is given in arxiv:1309.2484, i.e.

Article Deriving the time-dependent Schrodinger m- and p-equations f...

Phase has consequences in interfrrence

Or diffraction exprriments.